Our aim is to use the logic of public announcements and more generally dynamic epistemic logics as a logic of speech acts. To that end we start from a simple multimodal logic of beliefs and goals (without common belief), and add public announcements. We suppose that announcements do not modify goals. We then consider several variants of speech acts of assertive and directive force and provide a modeling in terms of speech acts

Andreas Herzig

Emiliano Lorini

Nadine Guiraud

Our aim is to use the logic of public announcements and more generally dynamic epistemic logics as a logic of speech acts. To that end we start from a simple multimodal logic of beliefs and goals (without common belief), and add public announcements. We suppose that announcements do not modify goals. We then consider several variants of speech acts of assertive and directive force and provide a modelling in terms of speech acts

Guiraud, Nadine

Herzig, Andreas

Lorini, Emiliano

Dagstuhl Research Online Publication Server

Speech acts as announcementsNadine Guiraud and Andreas Herzig and Emiliano LoriniUniversit Paul SabatierIRIT-LILaC118 route de NarbonneF-31062 Toulouse Cedex 9FranceAbstractOur aim is to use the logic of public announcements and moregenerally dynamic epistemic logics as a logic of speech acts.To that end we start from a simple multimodal logic of be-liefs and goals (without common belief), and add public an-nouncements. We suppose that announcements do not mod-ify goals. We then consider several variants of speech acts ofassertive and directive force and provide a modeling in termsof speech acts.1. IntroductionIn this paper, we consider the domain of communicationbetween agents, in particular we describe the dynamics ofmental states due to communication by means of speechacts. Starting from the representation of speech acts by(Grice 1989), we consider the mental states which definethe success preconditions of a speech act: intentions and be-lief. Thus Alice can say to Ben I like cookies. The mentalstate corresponding to this action, if Alice is honest, is thatAlice wants that Ben believes that she likes cookies. Thisis the well-known assertive speech act. On this way, Gricedefines other kinds of speech acts we consider in this paper:assert, inform , confirm , request, yes-no question. We donot only consider the mental states of the speaker, but alsowe represent the mental states of the hearer. In particular,we represent how the speech act influences beliefs and in-tentions of a locutor.In agreement with Grice’s theory, we suppose that whenan agent speaks, he informs about his mental states. In theprevious example, when Ben hears Alice’s utterance I likecookies, whether he thinks Alice is honest or not, he starts tobelieve that Alice wants him to beliefs that she likes cookies.Thus, in case of a successfull communication, the hearer willunderstand the mental states of the speaker.The success preconditions of speech acts are formalizedin this work by means of a modal logic of beliefs and goals.Similar formalisms have been used in (Herzig and Longin2002; david Sadek 2000). The logic is extended with modaloperators for announcements of Dynamic Epistemic Logic(DEL) (van Ditmarsch, van der Hoek, and Kooi 2007) inorder to model the dynamics speech acts. Our formalismThis paper has been presented at the LSIR workshop at IJCAI2009.enables to represent how a speech act of the speaker modi-fies the beliefs of the hearer and so to capture the dynamicdimension of a dialogue between agents.The paper is organized as follows: we first present thestatic modal logic of beliefs and goals. Then we provide adefinition of model update by an announcement. Finally, weformally characterize the success preconditions of severalkinds of speech acts and we develop an analysis of speechact dynamics.2. A logic of beliefs and goalsLet PRP = {p, q, . . .} be a countable set of propositionalletters, and let AGT = {i, j, . . .} be a countable set ofagents. The set of formulas of our logic of beliefs and goalsis defined by the following BNF:φ ::= p | > | ¬φ | φ ∧ φ | Bel iφ | Goal iφwhere p ranges over PRP and i ranges over AGT . Bel iφreads “agent i believes that φ” and Goal iφ reads “agent iwants that φ”. The others boolean operators⊥,∨,→ and↔are defined in the standard way.Models for that language are of the form M =〈W,B,G, V 〉 where• W is a nonempty set of possible worlds;• B : AGT −→ 2W×W associates a belief accessibilityrelation to every agent;• G : AGT −→ 2W×W associates a goal accessibility re-lation to every agent;• V : W −→ 2PRP associates to every possible world avaluation.We use Bi(w) to denote the set {w ′ | 〈w ,w ′〉 ∈ Bi}, Gi(w)to denote the set {w ′ | 〈w ,w ′〉 ∈ Gi}. Bi(w) is the set ofworlds that are compatible with agent i’s beliefs at world wand Gi(w) is the set of worlds that are compatible with agenti’s goals at world w .The truth conditions are as usual:• M,u |= Bel iφ iff M,v |= φ for every v such that uBiv;• M,u |= Goal iφ iff M,v |= φ for every v such that uGiv;The logic of that class of models is fairly standard. It canbe axiomatized by the principles of K45 for the belief op-erator Bel i, the principles of KD for the goal modal Goal i,Dagstuhl Seminar Proceedings 09351 Information processing, rational belief change and social interaction http://drops.dagstuhl.de/opus/volltexte/2009/2293plus the following axiom schemas of positive and negativeintrospection for goals:(1) Goal iφ→ Bel iGoal iφ(2) ¬Goal iφ→ Bel i¬Goal iφAxioms 4 and 5 for beliefs correspond together to the fol-lowing semantic constraint (C.1) on models, Axiom D forgoals corresponds to the constraint (C.2) and, finally, theaxioms of positive and negative introspection for goals cor-respond together to the semantic constraint (C.3).(C.1) w ′ ∈ Bi(w) implies Bi(w) = Bi(w ′)(C.2) Gi(w) 6= ∅(C.3) w ′ ∈ Bi(w) implies Gi(w) = Gi(w ′)3. Adding announcementsWe suppose that announcements are public, and that theyonly modify the agents’ beliefs, but not their goals. Tech-nically, our logic combines Kooi’s version of public an-nouncement logic (Kooi 2007) with Gerbrandy’s logic ofprivate updates (Gerbrandy 1999; Gerbrandy and Groen-eveld 1997). The latter allows us to update beliefs withoutupdating goals, and is a particular dynamic epistemic logic(Baltag, Moss, and Solecki 1998; Baltag and Moss 2004;van Ditmarsch, van der Hoek, and Kooi 2007).1We extend our language by modal operators of public an-nouncement by adding [φ!]φ to the above BNF. The formula[ψ!]φ reads “φ holds after the public announcement of ψ”.In order to give semantics to announcements we de-fine the update of a model by an announcement. The up-date of M = 〈W,B,G, V 〉 by φ! is the model Mφ! =〈Wφ!,Bφ!,Gφ!, V φ!〉 where• Wφ! = {ub : u ∈W} ∪ {ug : u ∈W};• Bφ! = {〈ub, vb〉 : 〈u, v〉 ∈ B and M, v |= φ}∪{〈ug, vg〉 :〈u, v〉 ∈ B};• Gφ! = {〈ub, vg〉 : 〈u, v〉 ∈ G} ∪ {〈ug, vg〉 : 〈u, v〉 ∈ G};• V φ!(ub) = V φ!(ug) = V (u).Basically, the effect of an announcement is to shrink the setof belief accessible worlds, while keeping constant the set ofgoal accessible worlds.Proposition 1. For every φ, if M is a model of our logic ofbeliefs and goals then Mφ! is still a model of our logic ofbeliefs and goals.Proof. The proof consists in verifying the conservation ofsemantic properties seen ahead.• For C.1: There are two cases for belief after announce-ment: either w2b ∈ Bψ!i (w1b) or w2g ∈ Bψ!i (w1g). In eachcase, it implies by definition that w2 ∈ Bi(w1). ThenBi(w1) = Bi(w2), because M satisfies constraint C.1.And also assume w3 ∈ Bi(w2). Then w3 ∈ Bi(w1).1For readers who are familiar with these logics: our event mod-els have two possible events b and g, where Bi = {〈b, b〉, 〈g, g〉}and Gi = {〈b, g〉, 〈g, g〉}.In case w2b ∈ Bψ!i (w1b): if w3 |= ψ then w3b ∈Bψ!i (w2b) and w3b ∈ Bψ!i (w1b). And if w3b 6|= ψthen w3b 6∈ Bψ!i (w2b) and w3b 6∈ Bψ!i (w1b) and w3g 6∈Bψ!i (w2b) and w3g 6∈ Bψ!i (w1b).Then Bψ!i (w1b) = Bψ!i (w2b).In case w2g ∈ Bψ!i (w1g): whether or not w3 |= ψ thenw3g ∈ Bψ!i (w2g) and w3g ∈ Bψ!i (w1g).Then Bψ!i (w1g) = Bψ!i (w2g).Therefore, M ψ! satisfies constraint C.1.• For C.2: Let us suppose Gψ!i (wb) = ∅ . Then Gi(w) = ∅by the semantic definition, and it is contradictory with theconstraints on M . Then Gψ!i (wb) 6= ∅. It is the same prooffor Gψ!i (wg) 6= ∅. Therefore M ψ! satisfies constraint C.2.• For C.3: There are two cases for belief after announce-ment: either w2b ∈ Bψ!i (w1b) or w2g ∈ Bψ!i (w1g). In eachcase, it implies by definition that w2 ∈ Bi(w1). ThenGi(w1) = Gi(w2), because M satisfies constraint C.3.And also assume w3 ∈ Gi(w2). Then w3 ∈ Gi(w1).In case w2b ∈ Bψ!i (w1b) : w3g ∈ Gψ!i (w2b) and w3g ∈Gψ!i (w1b). And in case w2g ∈ Bψ!i (w1g) : w3g ∈ Gψ!i (w2g)and w3g ∈ Gψ!i (w1g).Then Gψ!i (w1) = Gψ!i (w2) in both cases.Therefore, M ψ! satisfies constraint C.3.Items 1, 2 and 3 together imply that M ψ! is a model of ourlogic of belief and goal.The truth condition for announcement is:• M,u |= [ψ!]φ iff Mψ!, ub |= φWe suppose the usual definitions of validity and satisfia-bility. Then the following equivalences are valid:(3) [ψ!]p↔ p if p ∈ PRP(4) [ψ!]¬φ↔ ¬[ψ!]φ(5) [ψ!](φ ∧ χ)↔ ([ψ!]φ ∧ [ψ!]χ)(6) [ψ!]Bel iφ↔ Bel i(ψ → [ψ!]φ)(7) [ψ!]Goal iφ↔ Goal iφProof.(3) 〈W,B,G, V 〉,w |= [ψ!]piff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= piff w ∈ V ψ!(p)iff w ∈ V (p)iff 〈W,B,G, V 〉,w |= p.(4) 〈W,B,G, V 〉,w |= [ψ!]¬ϕiff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ¬ϕiff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb 6|= ϕiff 〈W,B,G, V 〉,w 6|= [ψ!]ϕiff 〈W,B,G, V 〉,w |= ¬[ψ!]ϕ.(5) 〈W,B,G, V 〉,w |= [ψ!](ϕ1 ∧ ϕ2)iff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ϕ1 ∧ ϕ2iff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ϕ1and 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ϕ2iff 〈W,B,G, V 〉,w |= [ψ!]ϕ1 and 〈W,B,G, V 〉,w |= [ψ!]ϕ2iff 〈W,B,G, V 〉,w |= [ψ!]ϕ1 ∧ [ψ!]ϕ2.(6) We show that the equivalent formula¬[ψ!]Bel iϕ↔ ¬Bel i(ψ → [ψ!]ϕ)is valid:〈W,B,G, V 〉,w |= ¬[ψ!]Bel iϕiff 〈W,B,G, V 〉,w |= [ψ!]¬Bel iϕ, by (4),iff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ¬Bel iϕiff ∃w ′b ∈ Bψ!(wb) s.t. 〈Wψ!,Bψ!,Gψ!, V ψ!〉,w ′b |= ¬ϕiff ∃w ′ ∈ B(w) s.t. 〈W,B,G, V 〉,w ′ |= ψ and〈W,B,G, V 〉,w ′ |= [ψ!]¬ϕiff ∃w ′ ∈ B(w) s.t. 〈W,B,G, V 〉,w ′ |= ψ ∧¬[ψ!]ϕ , by (4),iff ∃w ′ ∈ B(w) s.t. 〈W,B,G, V 〉,w ′ |= ¬(ψ → [ψ!]ϕ)iff 〈W,B,G, V 〉,w |= ¬Bel i(ψ → [ψ!]ϕ).(7) We show that the equivalent formula¬[ψ!]Goal iϕ↔ ¬Goal iϕis valid:〈W,B,G, V 〉,w |= ¬[ψ!]Goal iϕiff 〈W,B,G, V 〉,w |= [ψ!]¬Goal iϕ, by (4),iff 〈Wψ!,Bψ!,Gψ!, V ψ!〉,wb |= ¬Goal iϕiff ∃w ′g ∈ Gψ!(wb) s.t. 〈Wψ!,Bψ!,Gψ!, V ψ!〉,w ′g |= ¬ϕiff ∃w ′ ∈ G(w) s.t. 〈W,B,G, V 〉,w ′ |= ¬ϕiff 〈W,B,G, V 〉,w |= ¬Goal iϕ.The following rules of equivalence for announcements(8) from φ↔ φ′ infer [ψ!]φ↔ [ψ!]φ′(9) from ψ ↔ ψ′ infer [ψ!]φ↔ [ψ′!]φpreserve validity.The above equivalences 3-7 and rules of inference 8-9 areso-called reduction axioms: they allow to ‘move’ announce-ment operators across the other logical operators ‘inside’formulas, and to eliminate them once they face a propo-sitional letters. Rewriting formulas starting from the in-nermost announcements, we can in this way eliminate an-nouncement operators from formulas, resulting in an equiv-alent formula of the underlying logic of beliefs and goals.This provides a completeness result for our logic of an-nouncements.Theorem 1. Our logic of beliefs and goals is sound andcomplete.4. Speech acts as announcementsIn dynamic epistemic logics announcements are usuallyviewed as communication actions performed by an agentthat is ‘outside the system’, i.e. that is not part of the setof agents AGT under consideration. Our aim is to investi-gate how communication actions performed by agents fromAGT can be modelled in our logic of announcements. Tothat end we consider a particular subset of announcements:we say that formulas of the form Bel iψ! or Goal iψ! areabout the mental state of agent i. We identify announce-ments about the mental state of i, with actions performed byi. We follow Grice’s idea that speech acts are expressions ofintentions by the speaker, and consider that such expressionstake the form Goal iψ! of announcements of goals. Such an-nouncements of goals are directed towards an addressee.4.1 Assertive speech actsWe consider the basic assertive as the primitive speech actand we suppose that the other kinds of speech acts are de-fined from this primitive speech act. We identify the ex-pression “i asserts to j that ψ” with the announcement ofGoal iBel jψ:Assert i,jψdef= Goal iBel jψ!One might express a more complex notion of as-sertive speech act whose content is communicative in-tention in the sense of Grice (Grice 1989) of the formGoal iBel jGoal iBel jψ, i.e. agent i wants that j believesthat i wants that j believes ψ. In the classical Gricean viewof linguistic communication, a communicative intention ofthe speaker is aimed at the recognition by the hearer of thespeaker’s goal of making the hearer to believe something.Starting from the basic assertive, we define the two speechacts inform and confirm. The speech act inform is a speechact assert under the precondition that i believes that j has noopinion about ψ. We identify it therefore with the announce-ment of Bel i(¬Bel jψ ∧ ¬Bel j¬ψ) ∧Goal iBel jψ:Informi,jψdef= Bel i(¬Bel jψ ∧¬Bel j¬ψ)∧Goal iBel jψ!The speech act confirm is a speech act assert under theprecondition that i believes that j believes ψ. We iden-tify it therefore with the announcement of Bel iBel jψ ∧Goal iBel jψ:Confirmi,jψdef= Bel iBel jψ ∧Goal iBel jψ!In both cases one might add the sincerity condition of as-sertives to the announcement: the speaker believes what heasserts. We then would write Bel iψ∧Goal iBel jψ instead ofGoal iBel jψ for the basic assertive of the form Assert i,jϕ,etc.4.2 Directive speech actsThe speech act request is the basic speech act of the directivekind. We define it from the assertive. We identify “i requestsj that ψ” with “i asserts that he wants that j has the goal thatψ”:Request i,jψdef= Assert i,jGoal iGoal jψThe yes-no query “i asks j whether ψ” is a particular caseof a request:Ask i,jψdef= Request i,j(Bel iψ ∨ Bel i¬ψ)Note that if we had the sincerity condition Bel iψ for theassertive speech act, under the condition ¬Bel i⊥, the actof request would imply Goal iGoal jψ and the yes-no querywould imply Goal iGoal j(Bel iψ ∨ Bel i¬ψ).Suppose j is cooperative. This could be expressed by theformulaBel jGoal iBel jGoal iGoal jp→ Goal jpUnder this hypothesis the satisfaction conditions can beguaranteed, i.e. we haveTheorem 2.` [Request i,jp](Bel jGoal iBel jGoal iGoal jp→ Goal jp)→ [Request i,jp]Goal jpProof. We have the theorems that` [Request i,jp]Bel jGoal iBel jGoal iGoal jpand ` [Request i,jp]Goal jp↔ Goal jpso by K `[Request i,jp](Bel jGoal iBel jGoal iGoal jp→ Goal jp)→ [Request i,jp]Goal jp4.3 Moore sentences in speech actsIn our logic one can study Moore-like sentences for beliefs.We can prove that the assertion of p ∧ ¬Bel ip leads toinconsistencies of the (higher-order) beliefs:Theorem 3.` Bel i(p ∧ ¬Bel ip)↔ Bel i⊥` [Assert i,jBel i(p ∧ ¬Bel ip)!]Bel jGoal iBel jBel i⊥Proof.• ` ¬Bel ip→ Bel i¬Bel ip (axiom 5)so ` ¬Bel i¬Bel ip→ Bel ipand ` Bel iBel ip ∧ ¬Bel i⊥ → ¬Bel i¬Bel ipso ` Bel iBel ip ∧ ¬Bel i⊥ → Bel ipand so ` Bel iBel ip ∧ ¬Bel i⊥ ↔ Bel ip ∧ ¬Bel i⊥Bel i(p ∧ ¬Bel ip)↔ Bel ip ∧ Bel i¬Bel ip ∧ (Bel i⊥ ∨ ¬Bel i⊥)↔ (Bel ip∧Bel i¬Bel ip∧Bel i⊥)∨(Bel ip∧Bel i¬Bel ip∧¬Bel i⊥)↔ Bel i(p ∧ ¬Bel ip ∧ ⊥) ∨ (Bel iBel ip ∧ ¬Bel i⊥ ∧Bel i¬Bel ip)↔ Bel i⊥Therefore ` Bel i(p ∧ ¬Bel ip)↔ Bel i⊥• So [Assert i,jBel i(p ∧ ¬Bel ip)!]Bel jGoal iBel jBel i⊥iff Bel j(Goal iBel jBel i(p ∧ ¬Bel ip)→[Goal iBel jBel i(p ∧ ¬Bel ip)!]Goal iBel jBel i⊥) by (6)iff Bel j(Goal iBel jBel i(p ∧ ¬Bel ip)→Goal iBel jBel i⊥) by (7)iff Bel j(Goal iBel jBel i⊥ → Goal iBel jBel i⊥)iff Bel j>which is a tautology.Therefore` [Assert i,jBel i(p ∧ ¬Bel ip)!]Bel jGoal iBel jBel i⊥On the same way, we can prove that` [Assert i,jBel i(p ∧ ¬Bel ip)!]Bel iGoal iBel jBel i⊥5. Discussion and conclusionAn immediate extension of our logic is to add announce-ments modifying goals, similar to (van Benthem 2007;Roy 2008; van Benthem, Girard, and Roy to appear). Suchannouncements would leave an agent’s beliefs unaltered. Inorder to model this form of goal change, a product construc-tion symmetric to the one for beliefs can be used.Another issue we intend to investigate in the future is thetheory of speech acts in the context of indirected commu-nication. The idea is to extend the analysis presented inthis paper to the case in which agents make assertions, re-quests, queries, etc. to other agents by using a shared black-board which can be observed by other agents connected toit. In this case, an agent’s speech act (defined in terms of an-nouncements) modify and update the contents of the black-board which is accessible to other agents, rather than directlymodifying the beliefs and goals of other agents. In order tomodel this notion of blackboard, as a form of artefact usedfor mediating communication between agents, we intend touse the logic of acceptance recently proposed in (Lorini etal. 2009).We also leave open the problem of integrating belief revi-sion mechanisms into our logic, and refer to (Aucher 2003;2008; van Ditmarsch, van der Hoek, and Kooi 2007; Baltagand Smets 2007) for possible solutions.ReferencesAucher, G. 2003. A combined system for update logic andbelief revision. Master’s thesis, University of Amsterdam.Aucher, G. 2008. Perspectives on belief and change. Ph.D.Dissertation, Universite´ de Toulouse.Baltag, A., and Moss, L. S. 2004. Logics for epistemicprograms. Synthese 139(2):165–224.Baltag, A., and Smets, S. 2007. From conditional proba-bility to the logic of doxastic actions. In Proc. TARK 2007,52–61. Presses Universitaires de Louvain.Baltag, A.; Moss, L. S.; and Solecki, S. 1998. The logicof public announcements, common knowledge, and privatesuspicions. In Proc. TARK’98, 43–56. Morgan Kaufmann.david Sadek. 2000. Dialogue acts are rational plans. InMartin M. Taylor, F. N., and Bouwhuis, D. G., eds., TheStructure of Multimodal Dialogue II, 167–187. Amsterdam/ Philadelphia: John Benjamins.Gerbrandy, J., and Groeneveld, W. 1997. Reasoning aboutinformation change. J. of Logic, Language and Information6(2).Gerbrandy, J. 1999. Bisimulations on Planet Kripke. Ph.D.Dissertation, University of Amsterdam.Grice, H. P. 1989. Studies in the way of words. USA:Harvard University Press, 3rd edition.Herzig, A., and Longin, D. 2002. A logic of intention withcooperation principles and with assertive speech acts ascommunication primitives. In Castelfranchi, C., and John-son, W. L., eds., Proc. of the first Int. Joint Conf. on Au-tonomous Agent and Multi-Agent System (AAMAS 2002),Bologna, Italy, July 15–19, volume 2, 920–927. ACMPress.Kooi, B. 2007. Expressivity and completeness for publicupdate logic via reduction axioms. Journal of Applied Non-Classical Logics 17(2):231–253.Lorini, E.; Longin, D.; Gaudou, B.; and Herzig, A.2009. The logic of acceptance: grounding institutions onagents’ attitudes. Journal of Logic and Computation DOI:10.1093/logcom/exn103.Roy, O. 2008. Thinking before acting: intentions, logic,rational choice. Ph.D. Dissertation, University of Amster-dam, ILLC dissertation series.van Benthem, J.; Girard, P.; and Roy, O. to appear. Every-thing else being equal: A modal logic for ceteris paribuspreferences. Journal of Philosophical Logic.van Benthem, J. 2007. Dynamic logic of prefer-ence upgrade. Journal of Applied Non-Classical Logics17(2):157–182.van Ditmarsch, H. P.; van der Hoek, W.; and Kooi, B. 2007.Dynamic Epistemic Logic. Kluwer Academic Publishers.

Speech acts as announcements

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